Calculating div(theta) and tangent curves

[renegade05]

Homework Statement

Calculate ∇Θ where $$Θ(x)=\frac{\vec{p} \cdot \vec{x}}{r^3}$$. Here $$\vec{p}$$ is a constant vector and $$r=|\vec{x}|$$. In addition, sketch the tangent curves of the vector function ∇Θ for $$\vec{p}=p\hat{z}$$

(b) Calculate $$∇ (cross) A → \vec{A}=\frac{\vec{m}x\vec{X}}{r^3}$$ m is constant vector. Sketch the tangent curves of ∇(cross)A for $$\vec{m}=m\hat{z}$$

Homework Equations

gradient vector

The Attempt at a Solution

Well when I apply the gradient vector to the function Θ I get many terms and a very ugly answer. I am not sure if this will clean up nicely? Is there an easier way of doing this then brute force? Also, I am not sure how to represent the tangent curve of the vector function $$\vec{p}=p\hat{z}$$ or $$\vec{m}=m\hat{z}$$.

 

[vanhees71]

Sometimes the Ricci calculus is easier than the nabla calculus. BTW: Both questions are not about div but about grad and curl.

For the first problem you have to calculate (Einstein summation convention implied)

$$\partial_j \Theta=\partial_j \left ( \frac{p_k x_k}{r^3} \right ).$$
This is now just the task to take the partial derivatives using the usual rules for differentiation (product rule in this case).

For the second problem note that
$$(\vec{\nabla} \times \vec{A})_j = \epsilon_{jkl} \partial_k A_l,$$
where $\epsilon_{jkl}$ is the Levi-Civita symbol.